RD Sharma 2019 Solutions for Class 7 Math Chapter 15 Properties Of Triangles are provided here with simple step-by-step explanations. These solutions for Properties Of Triangles are extremely popular among class 7 students for Math Properties Of Triangles Solutions come handy for quickly completing your homework and preparing for exams. All questions and answers from the RD Sharma 2019 Book of class 7 Math Chapter 15 are provided here for you for free. You will also love the ad-free experience on Meritnation’s RD Sharma 2019 Solutions. All RD Sharma 2019 Solutions for class 7 Math are prepared by experts and are 100% accurate.

Page No 15.12:

Question 1:

Two angles of a triangle are of measures 105° and 30°. Find the measure of the third angle.

Page No 15.12:

Question 10:

In each of the following, the measures of three angles are given. State in which cases, the angles can possibly be those of a triangle:
(i) 63°, 37°, 80°
(ii) 45°, 61°, 73°
(iii) 59°, 72°, 61°
(iv) 45°, 45°, 90°
(v) 30°, 20°, 125°

Answer:

(i)Weknowthatthesumofallthethreeanglesofatriangleisequalto180°.Now,letusfindthesumof63°,37°and80°.63°+37°+80°=180°Thesumof63°,37°and80°isequalto180°.Hence,wecansaythatthegivenanglescanbethoseofatriangle.(ii)Weknowthatthesumofallthethreeanglesofatriangleisequalto180°.Now,letusfindthesumof45°,61°and73°.45°+61°+73°=179°Thesumof45°,61°and73°isnotequalto180°.Hence,wecansaythatthegivenanglescannotbethoseofatriangle.(iii)Weknowthatthesumofallthethreeanglesofatriangleisequalto180°.Now,letusfindthesumof59°,72°and61°.59°+72°+61°=192°Thesumof59°,72°and61°isnotequalto180°.Hence,wecansaythatthegivenanglescannotbethoseofatriangle.(iv)Weknowthatthesumofallthethreeanglesofatriangleisequalto180°.Now,letusfindthesumof45°,45°and90°.45°+45°+90°=180°Thesumof45°,45°and90°isequalto180°.Hence,wecansaythatthegivenanglescanbethoseofatriangle.(v)Weknowthesumofallthethreeanglesofatraingleisequalto180°.Now,letusfindthesumof30°,20°and125°.30°+20°+125°=175°Thesumof30°,20°and125°isnotequalto180°.Hence,wecansaythatthegivenanglescannotbethoseofatriangle.
Therefore, we can conclude that in (i) and (iv), the angles can be those of a triangle.

Page No 15.12:

Question 11:

The angles of a triangle are in the ratio 3 : 4 : 5. Find the smallest angle.

Page No 15.13:

Question 19:

Is it possible to have a triangle, in which
(i) two of the angles are right?
(ii) two of the angles are obtuse?
(iii) two of the angles are acute?
(iv) each angle is less than 60°?
(v) each angle is greater than 60°?
(vi) each angle is equal to 60°?
Give reasons in support of your answer in each case.

Answer:

(i) No, because if there are two right angles in a triangle, then the third angle of the triangle must be zero, which is not possible.
(ii) No, because as we know that the sum of all three angles of a triangle is always 180°. If there are two obtuse angles, then their sum will be more than 180°, which is not possible in case of a triangle.
(iii) Yes, in right triangles and acute triangles, it is possible to have two acute angles.
(iv) No, because if each angle is less than 60°, then the sum of all three angles will be less than 180°, which is not possible in case of a triangle.
Proof :Letthethreeanglesofthetrianglebe∠A,∠Band∠C.Asperthegiveninformation,∠A<60°...(i)∠B<60°...(ii)∠C<60°...(iii)Onadding(i),(ii)and(iii),weget:∠A+∠B+∠C<60°+60°+60°∠A+∠B+∠C<180°Wecanseethatthesumofallthreeanglesislessthan180°,whichisnotpossibleforatriangle.Hence,wecansaythatitisnotpossibleforeachangleofatriangletobelessthan60°.

(v) No, because if each angle is greater than 60°, then the sum of all three angles will be greater than 180°, which is not possible.
Proof :Letthethreeanglesofthegiventrianglebe∠A,∠Band∠C.Asperthegiveninformation,∠A>60°...(i)∠B>60°...(ii)∠C>60°...(iii)Onadding(i),(ii)and(iii),weget:∠A+∠B+∠C>60°+60°+60°∠A+∠B+∠C>180°Wecanseethatthesumofallthreeanglesofthegiventrianglearegreaterthan180°,whichisnotpossibleforatriangle.Hence,wecansaythatitisnotpossibleforeachangleofatriangletobegreaterthan60°.

(vi) Yes, if each angle of the triangle is equal to 60°, then the sum of all three angles will be 180°, which is possible in case of a triangle.
Proof :Letthethreeanglesofthegiventrianglebe∠A,∠Band∠C.Asperthegiveninformation,∠A=60°...(i)∠B=60°...(ii)∠C=60°...(iii)Onadding(i),(ii)and(iii),weget:∠A+∠B+∠C=60°+60°+60°∠A+∠B+∠C=180°Wecanseethatthesumofallthreeanglesofthegiventriangleisequalto180°,whichispossibleincaseofatriangle.Hence,wecansaythatitispossibleforeachangleofatriangletobeequalto60°.

Page No 15.19:

Question 1:

In Fig., ∠CBX is an exterior angle of âABC at B. Name
(i) the interior adjacent angle
(ii) the interior opposite angles to exterior ∠CBX.
Also, name the interior opposite angles to an exterior angle at A.

Page No 15.21:

Question 15:

Explain the concept of interior and exterior angles and in each of the figures given below, find x and y.

Answer:

The interior angles of a triangle are the three angle elements inside the triangle.
The exterior angles are formed by extending the sides of a triangle, and if the side of a triangle is produced, the exterior angle so formed is equal to the sum of the two interior opposite angles.

Page No 15.24:

Question 1:

In each of the following, there are three positive numbers. State if these numbers could possibly be the lengths of the sides of a triangle:
(i) 5, 7, 9
(ii) 2, 10, 15
(iii) 3, 4, 5
(iv) 2, 5, 7
(v) 5, 8, 20

Answer:

(i) Yes, these numbers can be the lengths of the sides of a triangle because the sum of any two sides of a triangle is always greater than the third side.
Here,5+7>9,5+9>7,9+7>5

(ii) No, these numbers cannot be the lengths of the sides of a triangle becauseâ the sum of any two sides of a triangle is always greater than the third side, which is not true in this case.

(iii) Yes, these numbers can be the lengths of the sides of a triangle because the sum of any two sides of triangle is always greater than the third side.
Here,3+4>5,3+5>4,4+5>3

(iv) No, these numbers cannot be the lengths of the sides of a triangle becauseâ the sum of any two sides of a triangle is always greater than the third side, which is not true in this case.
Here,2+5=7

(v) No, these numbers cannot be the lengths of the sides of a triangle becauseâ the sum of any two sides of a triangle is always greater than the third side, which is not true in this case.â
Here,5+8<20

Page No 15.24:

Question 2:

In Fig., P is the point on the side BC. Complete each of the following statements using symbol ' = ', '>' or '<' so as to make it true:
(i) AP ... AB + BP
(ii) AP .... AC + PC
(iii) AP....12(AB+AC+BC)

Answer:

(i) In triangle APB, AP < AB + BP because the sum of any two sides of a triangle is greater than the third side.

(ii) In triangle APC, AP < AC + PC because the sum of any two sides of a triangle is greater than the third side.

(iii) AP <12(AB+AC+BC)
In triangles ABP and ACP, we can see that:
AP < AB + BP ...(i) (Because the sum of any two sides of a triangle is greater than the third side)
AP < AC + PC ...(ii) (Because the sum of any two sides of a triangle is greater than the third side)

Page No 15.25:

Question 5:

In âABC, ∠A = 100°, ∠B = 30°, ∠C = 50°. Name the smallest and the largest sides of the triangle.

Answer:

Because the smallest side is always opposite to the smallest angle, which in this case is 30o, it is AC.
Also, because the largest side is always opposite to the largest angle, which in this case is 100o, it is BC.

Page No 15.30:

Question 1:

State Pythagoras theorem and its converse.

Answer:

The Pythagoras Theorem: In a right triangle, the square of the hypotenuse is always equal to the sum of the squares of the other two sides.

Converse of the Pythagoras Theorem: If the square of one side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle, with the angle opposite to the first side as right angle.

Page No 15.30:

Question 7:

Two poles of heights 6 m and 11 m stand on a plane ground. If the distance between their feet is 12 m, find the distance between their tops.
[Hint: Find the hypotenuse of a right triangle having the sides (11 − 6) m = 5 m and 12 m]

Answer:

Page No 15.31:

Question 9:

The foot of a ladder is 6 m away from a wall and its top reaches a window 8 m above the ground. If the ladder is shifted in such a way that its foot is 8 m away from the wall, to what height does its top reach?

Question 15:

Answer:

Draw △ABC.
Draw a line BC = 3 cm.
At point C, draw a line at 105° angle with BC.
Take an arc of 4 cm from point C, which will cut the line at point A.
Now, join AB, which will be approximately 5.5 cm.AC2+BC2=42+32=9+16=25AB2=5.52=30.25

(AB)2≠ (AC)2 + (BC)2

Here,

(AB)2 > (AC)2 + (BC)2

Page No 15.31:

Question 16:

Answer:

First draw △ABC.
Draw a line BC = 3 cm.
At point C, draw a line at 80° angle with BC.
Take an arc of 4 cm from point C, which will cut the line at point A.
Now, join AB; it will be approximately 4.5 cm.AC2+BC2=42+32=9+16=25AB2=4.52=20.25

(AB)2≠ (AC)2 + (BC)2

Here,

(AB)2 < (AC)2 + (BC)2

Page No 15.31:

Question 1:

If the measures of the angles of a triangle are (2x)° , (3x − 5)° and (4x − 13)°. Then the value of x is
(a) 22
(b) 18
(c) 20
(d) 30

Page No 15.35:

Question 20:

Answer:

In the given figure,AE∥BDgiven⇒∠A+∠B=180∘co-interiorangles
Sum of all the angles of a polygon is given by n-2×180°, where n is the number of sides of the polygon.
The given polygon has number of sides(n) = 5
So, sum of all the angles = n-2×180°=5-2×180°=3×180°=540°⇒∠A+∠B+∠C+∠D+∠E=540°⇒180°+132°+84°+3x=540°since,∠A+∠B=180°⇒3x+396°=540°⇒3x=540°-396°=144°
Divide both sides by 3, we getx=48°

Page No 15.36:

Question 24:

Which of the following is the set of measures of the sides of a triangle?
(a) 8 cm, 4 cm, 20 cm
(b) 9 cm, 17 cm, 25 cm
(c) 11 cm, 16 cm, 28 cm
(d) None of these

Answer:

We knwno that Triangle Inequality Theorem states that the sum of two side lengths of a triangle is always greater than the third side.
Using this in (a), we get
8 + 4 â¯ 20
⇒ 12 â¯ 20
So, triangle is not possible
Using this in (b), we get
9 + 17 > 25
⇒ 26 > 25
and
9 + 25 > 7
⇒ 34 > 7
and
17 + 25 > 9
⇒ 42 > 9
So, triangle is possible.
Using this in (c), we get
11 + 16 â¯ 28
⇒ 27 â¯ 28
So, triangle is not possible.
Hence, the correct answer is option (b).

Page No 15.36:

Question 25:

Answer:

In (a)
122 + 52 = 132
⇒ 144 + 25 = 169
⇒ 169 = 169
Since, the sum of the square of two smallest side is equal to the square of largest side.
Hence, a right triangle can be constructed.

In (b)
82 + 62 = 102
⇒ 44 + 36 = 100
⇒ 100 = 100
Since, the sum of the square of two smallest side is equal to the square of largest side.
Hence, a right triangle can be constructed.

In (c)
52 + 92 ≠ 112
⇒ 25 + 81 ≠ 121
⇒ 106 ≠ 121
Since, the sum of the square of two smallest side is not equal to the square of largest side.
Hence, a right triangle can not be constructed.
Hence, the correct answer is option (c).

Page No 15.36:

Question 26:

Answer:

In (a)
32 + 42 = 52
⇒ 9 + 16 = 25
⇒ 25 = 25
Since, the sum of the square of two smallest number is equal to the square of largest number.
Hence, it is a Pythagorean triplet.
In (b)
82 + 152 = 172
⇒ 64 + 225 = 289
⇒ 289 = 289
Since, the sum of the square of two smallest number is equal to the square of largest number.
Hence, it is a Pythagorean triplet.
In (c)
72 + 242 = 252
⇒ 49 + 576 = 625
⇒ 625 = 625
Since, the sum of the square of two smallest number is equal to the square of largest number.
Hence, it is a Pythagorean triplet.
In (d)
132 + 262 ≠ 292
⇒ 169 + 676≠ 841
⇒ 845 ≠ 841
Since, the sum of the square of two smallest number is not equal to the square of largest number.
Hence, it is not a Pythagorean triplet.
Hence, the correct answer is option (d).

Page No 15.36:

Question 27:

In a right triangle, one of the acute angles is four times the other. Its measure is
(a) 68°
(b) 84°
(c) 80°
(d) 72°

Answer:

In (c)
BC2 = AC2 + AB2
⇒ (17)2 = (15)2 + (8)2
⇒ 289 =225 + 64
⇒ 289 = 289
Since, the sum of the square of two smallest side is equal to the square of largest side.
Hence, ABC is a right angle triangle at A.
Hence, the correct answer is option (c).

Page No 15.4:

Question 1:

Take three non-collinear points A, B and C on a page of your notebook. Join AB, BC and CA. What figure do you get? Name the triangle. Also, name
(i) the side opposite to ∠B
(ii) the angle opposite to side AB
(iii) the vertex opposite to side BC
(iv) the side opposite to vertex B.

Answer:

The figure that we get is that of a triangle.
The name of the triangle is △ABC.
(i) The side opposite to ∠Bis AC.
(ii) The angle opposite to AB is ∠ACB.
(iii) The vertex opposite to BC is A.
(iv) The side opposite to the vertex B is AC.

Page No 15.4:

Question 2:

Take three collinear points A, B and C on a page of your note book. Join AB, BC and CA. Is the figure a triangle? If not, why?

Answer:

No, the figure is not a triangle. By definition, a triangle is a plane figure formed by three non-parallel line segments.

Page No 15.4:

Question 3:

Distinguish between a triangle and its triangular region.

Answer:

A triangle is a plane figure formed by three non-parallel line segments, whereas, its triangular region includes the interior of the triangle along with the triangle itself.

Page No 15.4:

Question 4:

In Fig., D is a point on side BC of a âABC. AD is joined. Name all the triangles that you can observe in the figure. How many are they?

Answer:

We can observe the following three triangles in the given figure:
1. △ ABC
2. △ ACD
â3. â△ ADB

Page No 15.4:

Question 5:

In Fig., A, B, C and D are four points, and no three points are collinear. AC and BD intersect at O. There are eight triangles that you can observe. Name all the triangles.

Answer:

The eight triangles that can be observed in the given figure are as follows:1.△AOD2.△AOB3.△BOC4.△COD5.△ACD6.△ACB7.△ADB8.△CDB

Page No 15.4:

Question 6:

What is the difference between a triangle and triangular region?

Answer:

A triangle is a plane figure formed by three non-parallel line segments, whereas, a triangular region is the interior of a triangle along with the triangle itself.

Answer:

(i) A triangle is a plane figure formed by three non-parallel line segments.
(ii) The three sides and the three angles of a triangle are together known as the parts or elements of that triangle.
(iii) A scalene triangle is a triangle in which no two sides are equal.
(iv) An isosceles triangle is a triangle in which two sides are equal.
(v) An equilateral triangle is a triangle in which all three sides are equal.
(vi) An acute triangle is a triangle in which all the angles are acute (less than 90°).
(vii) A right angled triangle is a triangle in which one angle is right angled, i.e 90°.
(viii) An obtuse triangle is a triangle in which one angle is obtuse (more than 90°).
(ix) The interior of a triangle is made up of all such points that are enclosed within the triangle.
(x) The exterior of a triangle is made up of all such points that are not enclosed within the triangle.

Page No 15.4:

Question 8:

In Fig., the length (in cm) of each side has been indicated along the side. State for each triangle whether it is scalene, isosceles or equilateral:

Answer:

(i) This triangle is a scalene triangle because no two sides are equal.
(ii) This triangle is an isosceles triangle because two of its sides, viz. PQ and PR, are equal.
(iii) This triangle is an equilateral triangle because all its three sides are equal.
(iv) This triangle is a scalene triangle because no two sides are equal.
(v) This triangle is an isosceles triangle because two of its sides are equal.

Page No 15.5:

Question 9:

In Fig., there are five triangles. The measures of some of their angles have been indicated. State for each triangle whether it is acute, right or obtuse.

Answer:

(i) This is a right triangle because one of its angles is 90°.
(ii) This is an obtuse triangle because one of its angles is 120°, which is greater than 90°.
(iii) This is an acute triangle because all its angles are acute angles (less than 90°).
(iv) This is a right triangle because one of its angles is 90°.
(v) This is an obtuse triangle because one of its angles is 110°, which is greater than 90°.

Page No 15.6:

Question 10:

Fill in the blanks with the correct word/symbol to make it a true statement:
(i) A triangle has ....... sides.
(ii) A triangle has ....... vertices.
(iii) A triangle has ....... angles.
(iv) A triangle has ....... parts.
(v) A triangle whose no two sides are equal is known as .......
(vi) A triangle whose two sides are equal is known as .....
(vii) A triangle whose all the sides are equal is known as .......
(viii) A triangle whose one angle is a right angle is known as .......
(ix) A triangle whose all the angles are of measure less than 90° is known as .......
(x) A triangle whose one angle is more than 90° is known as ......

Answer:

(i) three
(ii) three
(iii) three
(iv) six (three sides + three angles)
(v) a scalene triangle
(vi) an isosceles triangle
(vii) an equilateral triangle
(viii) a right triangle
(ix) an acute triangle
(x) an obtuse triangle

Page No 15.6:

Question 11:

In each of the following, state if the statement is true (T) or false (F):
(i) A triangle has three sides.
(ii) A triangle may have four vertices.
(iii) Any three line-segments make up a triangle.
(iv) The interior of a triangle includes its vertices.
(v) The triangular region includes the vertices of the corresponding triangle.
(vi) The vertices of a triangle are three collinear points.
(vii) An equilateral triangle is isosceles also.
(viii) Every right triangle is scalene.
(ix) Each acute triangle is equilateral.
(x) No isosceles triangle is obtuse.

Answer:

(i) True.
(ii) False. A triangle has three vertices.
(iii) False. Any three non-parallel line segments can make up a triangle.
(iv) False. The interior of a triangle is the region enclosed by the triangle and the vertices are not enclosed by the triangle.
(v) True. The triangular region includes the interior region and the triangle itself.
(vi) False. The vertices of a triangle are three non-collinear points.
(vii) True. In an equilateral triangle, any two sides are equal.
(viii) False. A right triangle can also be an isosceles triangle.
(ix) False. Each acute triangle is not an equilateral triangle, but each equilateral triangle is an acute triangle.
(x) False. An isosceles triangle can be an obtuse triangle, a right triangle or an acute triangle.